December 3, 2019

Why Does Maths Work?

A summary of Richard Hamming’s talk The Unreasonable Effectiveness of Mathematics”. All errors and omissions are entirely my own. All good stuff is down to Richard Hamming.

Cristiano Ronaldo is about to kick a football. He takes a step, places his supporting leg next to the ball, and strikes it sweetly with his laces. It curls wickedly into the bottom corner of the goal. He has just performed a difficult feat, a remarkable piece of biomechanical coordination. And he made it look easy. But it can be made even easier. All of the information about his goal can be captured in a few lines of mathematics. With only one equation and a few numbers describing Ronaldo’s run-up, I can exactly model the trajectory of Ronaldo’s strike. By understanding the present, I can predict the future.

There is no doubt that mathematics is effective. Planes fly, computers compute, and guided missiles explode. Creating any of these things would not be possible without mathematics. We are led to question mathematics by virtue of its very usefulness. We know maths works. But why does it work?

When asking such an expansive question, it is less overwhelming to approach the answer obliquely, to try and sneak a peak at the truth through the backdoor. To start we’ll look at some of the characteristics of maths which make it so uniquely useful in accurately describing physical reality. With luck, we’ll happen upon insights which will lead us to some answers.

One striking observation is that not only is maths so useful, but much of the time, the same mathematics is effective in wildly different areas. Fourier analysis is useful in signal processing, and quantum mechanics. Linear algebra is useful in quantum mechanics, and machine learning.

Equally striking is that the simplest mathematics is among the most useful. The most basic geometric constructions, like lines and planes are useful approximations of reality, and are described by the simplest algebra. It is remarkable how much one can learn about the world with only knowledge of basic algebra and arithmetic.

One of the main qualities of mathematics is the extension, or generalisation, of concepts into new territory. For example, Cardano’s extension of arithmetic properties like square roots to negative numbers created the concept of complex numbers — an extended set of numbers which obeyed the same properties of arithmetic as the reals. There have been many such extensions of number systems, from the natural numbers, to algebraic numbers, to transcendental, to the reals, to the complex, and further onwards. These extensions have been accepted, after some initial resistance, because they appeal aesthetically and because they are useful.

To accommodate these extensions, the definitions and rules of older mathematics must change subtly. Furthermore, as the field of mathematics has developed, standards for rigorous proof have increased. These qualities combine to give an amazing result: many proofs of old theorems are false, and have to be re-derived, or supplemented with additional axioms. Yet the theorems themselves remain true!

How can this be?

Clearly, mathematics doesn’t proceed automatically from god-given axioms to theorems, via proofs. Instead, theorems are often assumed, and good definitions are made from which a proof can then follow. Or intuitive definitions are made, and then these definitions are combined, like legos, to make a more ornate mathematical structure. Interesting qualities of this structure can then be catalogued by theorems, which we’ve proved along the way. Either way, both approaches are fundamentally driven by human intuition, and not an intrinsic property of an abstract object called mathematics”. The shape of mathematics reflects our minds.

We have tried to make mathematics consistent and aesthetically pleasing. As a result, it has become inordinately useful in describing reality. But why?

Some partial explanations:

We see what we look for.

Consider Pythagoras — one of the earliest great scientists. He discovered that the world we inhabit locally has a geometry with an L^2 norm — commonly referred to as the Pythagorean theorem. His theorem did not follow from some natural, or empirically derived postulates. Instead the theorem came from Pythagoras’ experience, and the postulates were necessary assumptions to maintain consistency in the mathematics.

Another example is one of my favourites — Benford’s Law. It happens that many numbers, including physical constants, are much more likely to have a lower leading digit than a higher one. This seems to be a deep, unexplained physical law, but really it is just an artefact of how we use numbers.

A lot of what we see comes from the glasses we put on.” And mathematics are the most powerful lenses we have.

Much of the unreasonable effectiveness of mathematics is simply because maths is the tool we choose to use. It is an example of Wittgenstein’s ruler: Instead of using maths to measure the world, we are using the world to measure our maths. It is our tacit knowledge about the world that we embed within our mathematics that makes mathematics such a useful tool for describing the world.

We select the kind of maths we use.

For a given slice of reality, most mathematics will not do a good job of describing it. If maths is a set of tools designed by humans, we choose the best one for the task at hand. When it turned out scalars were inappropriate for describing forces, we invented vectors. When it turned out fields were too complicated to be described by mere vectors, we invented tensors. Mathematical innovation has always been driven by the the problem we are trying to solve, which is inextricably linked to human intuition.

Science answers comparatively few problems.

When an inquisitive child is in her why?” phase, she only has to ask a few questions before she is waiting for an answer, and you start to get annoyed. Science is limited to questions which can be falsified, and as a result is limited by current technology. Many of the most interesting questions cannot (currently) be falsified, so science, and by extension mathematics, has nothing to say about them.

The evolution of man provided the model.

Human intuition begat maths, maths did not beget human intuition. It is plausible that evolution selects for organisms which have better models of their environment than their peers. Basic chains of cause and effect might be selected for, and logically deducing effects from their causes is one of the enduring qualities of mathematics.

This explanation is shocking, but not unreasonable.

Perhaps we have evolved the capacity for mathematics because it is a good descriptor of reality.

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